#415611

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(9,10,12,14)(11,13,15,16)(21,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8)(11,15)(13,16)(17,19)(18,21)(20,23)(22,24);(1,3,4,7)(2,6,5,8)(10,14)(13,16)

Polycyclic presentation:

2^9|a^2=h,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264129, #72264492, #72264521
maximal normal subgroup of	#72264129, #72264492, #72264521

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322807);